Hyperbolic manifolds with convex boundary

in Inventiones Mathematicae (2006), 163(1), 109--169

A hyperbolic proof of Pascal's Theorem

in Mathematical Intelligencer (2021), 43(2), 130--133

We provide a simple proof of Pascal's Theorem on cyclic hexagons, as well as a generalization by Möbius, using hyperbolic geometry.

Hyperelastic Elastography in a Large-Scale Bayesian Inversion Setting

Scientific Conference (2015, July)

We consider the problem of recovering the material parameters of a hyperelastic material [1] in the Bayesian inversion setting. In the Bayesian setting we can extract the statistics associated with ... [more ▼]

We consider the problem of recovering the material parameters of a hyperelastic material [1] in the Bayesian inversion setting. In the Bayesian setting we can extract the statistics associated with various sources of uncertainty, including noise, experimental deficiencies and incomplete observations of the domain. This will allow medical practitioners to make superior diagnosis decisions when presented with a quantitative measure of uncertainty in the recovered parameters. On the assumption of a Gaussian additive noise model, a Gaussian prior and a linear forward model, the posterior distribution of the material parameters given the observations will also be Gaussian. To ensure that the assumption of a linear forward model is valid, and that the posterior is approximated sufficiently well by a Gaussian distribution, we place a limit on the strain regime in which our current methodology applies. We are developing MCMC methods for exploring the non-Gaussian statistics of the posterior distribution. In the linear case, the covariance matrix of the posterior distribution is then characterised by the inverse of the Hessian of the objective functional evaluated at its minimiser. To extract statistical information from the large and dense Hessian we perform a low-rank approximation of the Hessian [2]. The eigenvectors associated with the lowest eigenvalues are the directions in parameter space that are least constrained by the observations. We implement this work within the dolfin-adjoint [3] software package. We derive the MPI-parallel finite element discretisation of the forward, adjoint (1st and 2nd order), and tangent linear models using the high-level differentiation tools available within the FEniCS project. We show results demonstrating the effects of partial observations and poor experimental design on the reliability of the recovered parameters. [1] N. H. Gokhale, P. E. Barbone, and A. A. Oberai, “Solution of the nonlinear elasticity imaging inverse problem: the compressible case,” Inverse Problems, 10.1088/0266-5611/24/4/045010 [2] H. P. Flath, L. C. Wilcox, V. Akçelik, J. Hill, B. van Bloemen Waanders, and O. Ghattas, “Fast Algorithms for Bayesian Uncertainty Quantification in Large-Scale Linear Inverse Problems Based on Low-Rank Partial Hessian Approximations,” SIAM J. Sci. Comput., 10.1137/090780717 [3] P. Farrell, D. Ham, S. Funke, and M. Rognes, “Automated Derivation of the Adjoint of High-Level Transient Finite Element Programs,” SIAM J. Sci. Comput., 10.1137/120873558 [less ▲]

Hyperideal polyhedra in the 3-dimensional anti-de Sitter space

in Advances in Mathematics (2022), 404(Paper No. 108441, 61 pp),

We study hyperideal polyhedra in the 3-dimensional anti-de Sitter space AdS3, which are defined as the intersection of the projective model of AdS3 with a convex polyhedron in RP3 whose vertices are all ... [more ▼]

We study hyperideal polyhedra in the 3-dimensional anti-de Sitter space AdS3, which are defined as the intersection of the projective model of AdS3 with a convex polyhedron in RP3 whose vertices are all outside of AdS3 and whose edges all meet AdS3. We show that hyperideal polyhedra in AdS3 are uniquely determined by their combinatorics and dihedral angles, as well as by the induced metric on their boundary together with an additional combinatorial data, and describe the possible dihedral angles and the possible induced metrics on the boundary. [less ▲]

Hyperosmotic stress induces immediate-early gene expression in ventricular adult cardiomyocytes.

in Biochemical and biophysical research communications (1993), 194(2), 642-6

Mammalian cells possessing osmosensors have long been described in brain and kidney. The genetic basis of the response to hyperosmotic stress has been well characterized in prokaryotes. In contrast, the ... [more ▼]

Mammalian cells possessing osmosensors have long been described in brain and kidney. The genetic basis of the response to hyperosmotic stress has been well characterized in prokaryotes. In contrast, the genetic response of eukaryotic cells is poorly understood. Therefore we investigated the effect of hypertonic NaCl and sucrose solutions on the transcriptional activation of the immediate-early genes (IEGs) egr-1 and c-fos in isolated ventricular adult rat cardiomyocytes. We observed that even small increases in osmolarity to 315 +/- 5 mosmol/l and 370 +/- 8 mosmol/l by hypertonic NaCl solution resulted in dose-dependent induction of egr-1 (4-and 5-fold) and c-fos (3-and 4-fold), respectively. Hypertonic sucrose solution had the same effect on egr-1 and c-fos mRNA levels while increased sucrose concentration under isotonic conditions had no effect. Cardiomyocytes exposed to hypertonic media did not significantly shrink as shown by a cell length measurement. We conclude that isolated adult cardiomyocytes possess an osmoreceptor mechanism which is able to sense even slight changes in osmolarity and to translate these into a transcriptional response of the myocardial IEG program. [less ▲]

[Hypertensive heart disease. Pathophysiological foundation, diagnosis and therapy].

in Deutsche medizinische Wochenschrift (1946) (1990), 115(39), 1480-6

The HyperVerse - Concepts for a Federated and Torrent-Based '3D Web'

in International Journal of Advanced Media and Communication (2008), 2(4), 331350

HyperVerse: Simulation and Testbed Reconciled

in International Journal of Advanced Media and Communication (2010), 4(2), 167-181

Hypoelliptic non-homogeneous diffusions

Scientific Conference (2002, October 10)

Hypomorphic Sperner systems and non-reconstructible functions

in Order: A Journal on the Theory of Ordered Sets and its Applications (2014)

A reconstruction problem is formulated for Sperner systems, and infinite families of non-reconstructible Sperner systems are presented. This has an application to a reconstruction problem for functions of ... [more ▼]

A reconstruction problem is formulated for Sperner systems, and infinite families of non-reconstructible Sperner systems are presented. This has an application to a reconstruction problem for functions of several arguments and identification minors. Sperner systems being representations of certain monotone functions, infinite families of non-reconstructible functions are thus obtained. The clones of Boolean functions are completely classified in regard to reconstructibility. [less ▲]